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1. Solving Rational Equations and Inequalities 5-5 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

2. Answers #21-48 mult. of 3 p 571-572 21. x=2 • x=6 • x=-3/2; x=2 • No solution 33. x=6/17 • x=7;x=-5 39. x=4 • x=6; x=-1 • No solution 48. x=-6

3. –(x – 1) 1 1 1 1 5x – 2 + 4x(x – 2) x – 2 x2 4x x2 x – Warm Up Add or subtract. Identify any x-values for which the expression is undefined. 1. x ≠ 0 or 2 2. x ≠ 0

4. Objective Solve rational equations and inequalities.

5. Vocabulary rational equation extraneous solution rational inequality

6. d r A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined by using the equation t = , where r is the average rate of speed. This equation is a rational equation.

7. To solve a rational equation, start by multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. This step eliminates the denominators of the rational expression and results in an equation you can solve by using algebra.

8. An extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation. When you solve a rational equation, it is possible to get extraneous solutions. These values should be eliminated from the solution set. Always check your solutions by substituting them into the original equation.

9. x x x 2 2 2 2(x – 8) + 2(x – 8) = 2(x – 8) 2(x – 8) + 2(x – 8) = 2(x – 8) 2x – 5 2x – 5 2x – 5 11 11 11 x – 8 x – 8 x – 8 x – 8 x – 8 x – 8 Example 2B: Extraneous Solutions Solve each equation. + = Multiply each term by the LCD, 2(x – 8). Divide out common factors. 2(2x – 5) + x(x – 8) = 11(2) Simplify. Note that x ≠ 8. Use the Distributive Property. 4x – 10 + x2 – 8x = 22

10. Example 2B Continued x2 – 4x – 32 = 0 Write in standard form. (x – 8)(x + 4) = 0 Factor. x – 8 = 0 or x + 4 = 0 Apply the Zero Product Property. x = 8 or x = –4 Solve for x. The solution x = 8 us extraneous because it makes the denominator of the original equation equal to 0. The only solution is x = –4.

11. x x + = + – = 0. 2 2 2x – 5 2x – 5 11 11 x – 8 x – 8 x – 8 x – 8 Example 2B Continued Check Write as Graph the left side of the equation as Y1. Identify the values of x for which Y1 = 0. The graph intersects the x-axis only when x = –4. Therefore, x = –4 is the only solution.

12. A rational inequality is an inequality that contains one or more rational expressions. One way to solve rational inequalities is by using graphs and tables.

13. Solve ≤ 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = and Y2 = 3. x x x–6 x–6 Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities (9, 3) The graph of Y1 is at or below the graph of Y2 when x < 6 or when x ≥ 9. Vertical asymptote: x = 6

14. Example 5 Continued Use a table. The table shows that Y1 is undefined when x = 6 and that Y1 ≤ Y2 when x ≥ 9. The solution of the inequality is x < 6 or x ≥ 9.

15. Solve ≥ 4 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = and Y2 = 4. x x x–3 x–3 Check It Out! Example 5a (4, 4) The graph of Y1 is at or below the graph of Y2 when x < 3 or when x ≥ 4. Vertical asymptote: x = 3

16. Check It Out! Example 5a continued Use a table. The table shows that Y1 is undefined when x = 3 and that Y1 ≤ Y2 when x ≥ 4. The solution of the inequality is x < 3 or x ≥ 4.

17. You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative.

18. Solve ≤ 3 algebraically. 6 (x – 8) ≤ 3(x – 8) x – 8 6 x–8 Example 6: Solving Rational Inequalities Algebraically Case 1 LCD is positive. Step 1 Solve for x. Multiply by the LCD. 6 ≤ 3x – 24 Simplify. Note that x ≠ 8. 30 ≤ 3x Solve for x. 10 ≤ x Rewrite with the variable on the left. x ≥ 10

19. Solve ≤ 3 algebraically. 6 x–8 Example 6 Continued Step 2 Consider the sign of the LCD. x – 8 > 0 LCD is positive. x > 8 Solve for x. For Case 1, the solution must satisfy x ≥ 10 and x > 8, which simplifies to x ≥ 10.

20. Solve ≤ 3 algebraically. 6 (x – 8) ≥ 3(x – 8) x – 8 6 x–8 Example 6: Solving Rational Inequalities Algebraically Case 2 LCD is negative. Step 1 Solve for x. Multiply by the LCD. Reverse the inequality. 6 ≥ 3x – 24 Simplify. Note that x ≠ 8. 30 ≥ 3x Solve for x. 10 ≥ x Rewrite with the variable on the left. x ≤ 10

21. Solve ≤ 3 algebraically. 6 6 x–8 x–8 The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality ≤ 3 is x < 8 or x ≥ 10, or {x|x < 8  x ≥ 10}. Example 6 Continued Step 2 Consider the sign of the LCD. x – 8 ‹ 0 LCD is negative. x ‹ 8 Solve for x. For Case 2, the solution must satisfy x ≤ 10 and x < 8, which simplifies to x < 8.

22. Solve ≥ –4 algebraically. 1 1 2 2 6 (x – 2)≥ –4(x – 2) x – 2 6 x–2 ≤x x≥ You Try! Example 6a Case 1 LCD is positive. Step 1 Solve for x. Multiply by the LCD. 6 ≥ –4x + 8 Simplify. Note that x ≠ 2. –2 ≥ –4x Solve for x. Rewrite with the variable on the left.

23. Solve ≥ –4 algebraically. 1 2 For Case 1, the solution must satisfy and x > 2, which simplifies to x > 2. 6 x–2 x≥ You Try! Example 6a Continued Step 2 Consider the sign of the LCD. x – 2 > 0 LCD is positive. x > 2 Solve for x.

24. Solve ≥ –4 algebraically. 1 1 2 2 6 (x – 2)≤ –4(x – 2) x – 2 6 x–2 ≥x x≤ Check It Out! Example 6a Continued Case 2 LCD is negative. Step 1 Solve for x. Multiply by the LCD. Reverse the inequality. 6 ≤ –4x + 8 Simplify. Note that x ≠ 2. –2 ≤ –4x Solve for x. Rewrite with the variable on the left.

25. Solve ≥ –4 algebraically. 1 1 1 1 2 2 2 2 For Case 2, the solution must satisfy and x < 2, which simplifies to . 6 6 The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality ≥ –4 is x > 2 or x ≤ , or {x| x > 2}. x–2 x–2 x≤ x≤ x≤ Check It Out! Example 6a Continued Step 2 Consider the sign of the LCD. x – 2 < 0 LCD is negative. x < 2 Solve for x.

26. Solve < 6 algebraically. 3 3 2 2 9 (x + 3)< 6(x + 3) x + 3 9 x+ 3 – < x x> – Check It Out! Example 6b Case 1 LCD is positive. Step 1 Solve for x. Multiply by the LCD. 9 < 6x + 18 Simplify. Note that x ≠ –3. Solve for x. –9 < 6x Rewrite with the variable on the left.

27. Solve < 6 algebraically. 3 3 2 2 x>– x>– For Case 1, the solution must satisfy and x > –3, which simplifies to . 9 x+ 3 Check It Out! Example 6b Continued Step 2 Consider the sign of the LCD. x + 3 > 0 LCD is positive. x > –3 Solve for x.

28. Solve < 6 algebraically. 3 3 2 2 9 (x + 3)> 6(x + 3) x + 3 9 x+ 3 – > x x< – Check It Out! Example 6b Continued Case 2 LCD is negative. Step 1 Solve for x. Multiply by the LCD. Reverse the inequality. 9 > 6x + 18 Simplify. Note that x ≠ –3. –9 > 6x Solve for x. Rewrite with the variable on the left.

29. Solve < 6 algebraically. 3 3 3 2 2 2 For Case 2, the solution must satisfy and x < –3, which simplifies to x < –3. x<– 9 The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality < 6 is x < –3 or x > – , or {x| x < –3}. 9 x+ 3 x+ 3 x > – Check It Out! Example 6b Continued Step 2 Consider the sign of the LCD. x + 3 < 0 LCD is negative. x < –3 Solve for x.